Finite Group Databasebeta

← #26114108

Subgroups — #26114108

Order 12032.

614 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245986C211yesmaximal, characteristic
2#20243487C212yesmaximal, complemented
3#20241715C211yesmaximal, characteristic
4#20241240C212yesmaximal, complemented
5C47×#2482C211yesmaximal, characteristic
6#15689087C2212yes—
7#15689016—24no—
8#15688983—21no—
9#15688979—21no—
10C4×C22⋊Dic47C2212yes—
11#15688882—24no—
12#15688847—22no—
13C2×#117×C47C2211yescharacteristic
14C22×C22⋊C4×C47C2211yescharacteristic
15C47×#399C412yescomplemented
16C47×#397C412yescomplemented
17C47×#388—22no—
18C47×#385—21no—
19C47×#381C2211yescharacteristic
20C47×#379—21no—
21C2×#10625329—22no—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C4×Dic47D414yes—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C25×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47C2311yescharacteristic
34#117×C47—22no—
35#117×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47C2×C411yescharacteristic
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47C2×C412yes—
46C2×C22⋊C4×C47D412yes—
47C2×C42×C47—21no—
48#95×C47—22no—
49#95×C47—21no—
50#95×C47C2×C412yes—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—42no—
54C24×C47—42no—
55C24×C47—81no—
56C24×C47C22⋊C412yes—
57C24×C47C2×D411yescharacteristic
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—22no—
63C22×C4×C47—42no—
64C22×C4×C47—81no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D412yes—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4112yes—
70C22×Dic47—24no—
71C4⋊C4×C47—22no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—82no—
74C22⋊C4×C47—24no—
75C22⋊C4×C47—24no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—44no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—82no—
81C42×C47—42no—
82C22⋊Dic47—44no—
83C22⋊Dic47—82no—
84C22⋊Dic47—82no—
85C4⋊Dic47—82no—
86C4×Dic47—44no—
87#10625329—44no—
88C2×C47⋊C8—42no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—41no—
95C23×C47—42no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—81no—
102C23×C47—81no—
103C23×C47—88no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12011yescharacteristic
106C23×C47#12311yescharacteristic
107C2×C4×C47—24no—
108C2×C4×C47—22no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—81no—
122C2×C4×C47C4×D412yes—
123C2×C4×C47#12112yes—
124C2×Dic47—44no—
125C2×Dic47—44no—
126C2×Dic47—44no—
127C47⋊C8—82no—
128#9481—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—44no—
137C22×C47—44no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47—82no—
142C22×C47—84no—
143C22×C47—84no—
144C22×C47#38611yescharacteristic
145C22×C47#42112yes—
146C22×C47#45812yes—
147C4×C47—22no—
148C4×C47—22no—
149C4×C47—42no—
150C4×C47—82no—
151C4×C47—81no—
152C4×C47—81no—
153Dic47—84no—
154#2482D4711yescharacteristic
155#2312—471no—
156#1826—472no—
157#1462—471no—
158#1284—472no—
159C2×C47—22no—
160C2×C47—22no—
161C2×C47—41no—
162C2×C47—82no—
163C2×C47#189011yescharacteristic
164C2×C47#189711yescharacteristic
165C2×C47#191111yescharacteristic
166C2×#117C2×D4711yescharacteristic
167C22×C22⋊C4C2×D4711yescharacteristic
168#407—472no—
169#399Dic4712yescomplemented
170#397Dic4712yescomplemented
171#388—22no—
172#388—944no—
173#385—21no—
174#381C2×D4711yescharacteristic
175#380—941no—
176#379—21no—
177#379—941no—
178C4×C22⋊C4—472no—
179#339—944no—
180#323—942no—
181C47#948111yescharacteristic, complemented, Sylow
182C25C22⋊D4711yescharacteristic
183C23×C4C22⋊D4711yescharacteristic
184C23×C4C22×D4711yescharacteristic
185C2×M16—942no—
186#117—22no—
187#117—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—942no—
190C2×C4⋊C4C2×Dic4711yescharacteristic
191C2×C4⋊C4C2×Dic4711yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—942no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4C2×Dic4712yes—
202C2×C22⋊C4C22⋊D4712yes—
203C2×C42—21no—
204C2×C42—474no—
205C22⋊C8—942no—
206C22⋊C8—942no—
207#95—22no—
208#95—21no—
209#95—942no—
210#95—942no—
211#95C2×Dic4712yes—
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24C22⋊Dic4712yes—
218C24C2×C22⋊D4711yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4—944no—
227C22×C4C22⋊Dic4712yes—
228C22×C4D4×D4712yes—
229C22×C4#1062535212yes—
230C22×C4C22×Dic4711yescharacteristic
231C22×C4C2×C22⋊D4711yescharacteristic
232M16—1884no—
233C2×C8—1882no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—82no—
237C4⋊C4—3762no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—82no—
245C22⋊C4—3762no—
246C22⋊C4—3762no—
247C22⋊C4—1884no—
248C42—42no—
249C42—1884no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×C22⋊Dic4711yescharacteristic
266C23#1309399211yescharacteristic
267C23#1309400511yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—1884no—
284C2×C4—1884no—
285C2×C4—1884no—
286C2×C4D4×Dic4712yes—
287C2×C4#1309399612yes—
288C8—3762no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#1568900311yescharacteristic
305C22#1568914012yes—
306C22#1568931212yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—82no—
311C4—81no—
312C4—81no—
313C4—3764no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#2024380211yescharacteristic
319C2#2024383711yescharacteristic
320C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2482C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2482D47
hypocenterC47#9481
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20243487C2semidirect
4. #20241240C2semidirect
15. C47×#399C4semidirect
16. C47×#397C4semidirect
169. #399Dic47semidirect
170. #397Dic47semidirect
181. C47#9481semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202459862
#202434872
#202417152
#202412402
C47×#24822
#948147